Note: This is not a list of the official rules of any version of Dungeons and Dragons, it's just some ideas which can be used for any version of the game or any simplified version anyone might use.
The most common die used in table top games is the six sided die, known as d6. However, this is not the only type, as most Dungeons and Dragons players know well enough. In a dice set, one can generally find seven dice, as shown below.
Since the twenty sided die, known as the d20, is the one that is used most in Dungeons and Dragons, and people who play a simplified version of the game which only uses the d20, many players simply buy a d20 and that's it. So much so, that it's called the d20 system. Even so, the other dice can easily be introduced and used throughout the game. For instance, the d12, d10, d8 and d6 can be used to determine the amount of damage done depending on which weapon the player is using. For example, a two handed weapon can use a d12, a one handed axe or sword can use a d10, while daggers can use a d8. Recently, while we were playing a DnD campaign, the Dungeon Master decided to use the d6 for a new mechanic; when more enemies where on their way, he asked the players who can see farthest, for example elves, or illusion mages who might be able to see through an invisibility spell, to throw a d6, the resultant number being the number of enemies the player can see. Although some might not like this sort of mechanic, it's an idea one can use. After all, the point of a table top game is to be able to imagine and improvise. The d90 can be used in conjunction with the d10 to achieve any number between 1 and 100. This can be used for instances where the player is required to roll above a certain percentage; for example, to check whether a player can see through an illusion spell or not.
Also, the d20 can be used for things like initiative to see who gets to attack first, if such a mechanic is to be used, whether the player hits an enemy or not as well as to see if the player lands a critical hit or a critical miss. A critical hit (usually a two 20s in a row) deals extra damage while a critical miss (two 1s in a row) usually means you either drop your weapon or hit allies who happen to be nearby if you use a melee weapon, or allies in front of you if you're using ranged attacks (arrows, throwing knives, spells, etc.)
A note I would like to make before I close off is this; if you do not wish to purchase a whole dice set, perhaps when starting off the hobby in the first place, you can try out an app on your smartphone or tablet that simulates the use of dice, any number of sides, as shown below.
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